A268509 -id:A268509 - cp's OEIS frontend

A253181 Numbers n such that the distance between n^3 and the nearest square is less than n.

1, 2, 3, 4, 5, 9, 13, 15, 16, 17, 25, 32, 35, 36, 37, 40, 43, 46, 49, 52, 56, 63, 64, 65, 81, 99, 100, 101, 109, 121, 136, 143, 144, 145, 152, 158, 169, 175, 190, 195, 196, 197, 225, 243, 255, 256, 257, 289, 312, 317, 323, 324, 325, 331, 336, 351, 356, 361, 366, 377

Offset: 1

Alex Ratushnyak, Mar 23 2015

nonn

Distance can be zero, that is, cubes that are squares are included.

Numbers n such that A002938(n) < n.

			The distance between 5^3=125 and the nearest square 11^2=121 is less than 5, so 5 is in the sequence.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A002760.
Cf. A116885, A002938, A077119.
Cf. A268509, A268510.

dnsQ[n_]:=Module[{n3=n^3,sr},sr=Sqrt[n3];Min[n3-Floor[sr]^2, Ceiling[ sr]^2- n3]Harvey P. Dale, Dec 23 2015 *)

def isqrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s:  sr = s
        b>>=1
    return sr
for n in range(1000):
    cube = n*n*n
    r = isqrt(cube)
    sqr = r**2
    if cube-sqr < n or sqr+2*r+1-cube < n:  print(str(n), end=',')

A268510 Numbers x such that x^2 = y^3 + z (0 < abs(z) < y).

Original entry on oeis.org

3, 5, 11, 47, 58, 70, 181, 207, 225, 253, 282, 312, 375, 419, 500, 524, 985, 1015, 1138, 1586, 1710, 1746, 1874, 1986, 2315, 2619, 2723, 2765, 3788, 4072, 4120, 5511, 5644, 5805, 5859, 6022, 6159, 6576, 6717, 7002, 7320, 7970, 8030, 8669, 10615, 10681, 13252, 13537, 13788, 13860, 14113, 14404, 16725, 17537, 17615

Offset: 1

Daniel Mondot, Feb 06 2016

nonn

List of n such as n^2 is a near cube (see examples).
Numbers x such that x^2 = y^3 + 0 (e.g. 1000^2 = 100^3) are omitted.
Note that a delta of 17 appears often. See A029728.

			3^2 = 2^3 + 1
5^2 = 3^3 - 2
11^2 = 5^3 - 4
47^2 = 13^3 + 12
58^2 = 15^3 - 11
70^2 = 17^3 - 13
181^2 = 32^3 - 7
207^2 = 35^3 - 26
225^2 = 37^3 - 28
253^2 = 40^3 + 9
282^2 = 43^3 + 17
312^2 = 46^3 + 8
375^2 = 52^3 + 17
419^2 = 56^3 - 55
500^2 = 63^3 - 47
524^2 = 65^3 - 49
985^2 = 99^3 - 74

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A029728, A268509, A253181.

Select[Range@ 5000, Resolve@ Exists[{y, z}, And[Reduce[#^2 == (y^3 + z), {y, z}, Integers], 0 < Abs@ z < y]] &] (* Michael De Vlieger, Feb 07 2016 *)

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A253181 Numbers n such that the distance between n^3 and the nearest square is less than n.

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